The door switch on my very old Kenmore Heavy Duty dryer is broken. The metal tab that I assume activates the dryer has snapped off. Is there any way to start the machine manually without depending on that switch?

If you model has a visible switch, you can pull it out of the door, strip the ends, put a small wire nut on the wires and push it back into the opening where the switch was. You can also pull the timer panel off and jump the wires there. This should only be done as a temporary measure as you are defeating a safety feature.

That switch is a generic switch that you can get almost anywhere parts are sold for about $5.00. I would advise against cutting off the ends of the wires unless the ends were loose, which will cause it to heat up and ruin switch or cause a fire. You can buy replacement connectors that are insulated and will fit tight.